−∞ Lecture: On Number and Space

Isn’t it strange that 5 penguins, 5 apples, and 5 spoons have something in common, in spite of thewild differences between a penguin, an apple, and a spoon? An attempt to explain exactly what itis the collections share drives us inexorably towards the metaphysical question concerning the natureof a number. We will mostly avoid that question today. Scientists, on the whole, like to stay awayfrom questions of the ‘what is’ sort. They find it fruitful most of the time to concentrate rather on‘how,’ or, more precisely, on how something works. So a physicist may not be much better than the

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man in the street when called upon to define force, mass, position, or time, but he has a pretty goodunderstanding of relations like

md2x

dt2= F,

orF = G

mM

R2

telling him how a massive object moves, that is, changes position and velocity with time, when subjectto a force (gravity, for example). In a similar vein, we can, in pure mathematics, correctly performoperations like

5 + 3 = 8

or even1729× 691 = 1104739

without pausing overly long to consider the meaning of it all. Furthermore, whether the objectswe combine are penguins or apples, we are sure of the remarkable phenomenon that the abstractcalculation done once will allow us to use the result uniformly: five penguins when combined withthree penguins will definitely yield eight penguins, and the same for apples, spoons, or human beings.I believe some philosophers have attempted a definition to the effect that ‘five is what is common toall collections of five objects,’ or something equally unsatisfactory. Like it or not, it is reasonable thatthis definition encourages us to consider the number 5 as a variable of sorts, which can be assignedspecific values like 5 apples or 5 penguins in accordance with the needs of the situation. Schoolchildrenthese days are able to take the abstraction one step further, and casually consider equalities like

(x+ y)2 = x2 + 2xy + y2.

The meaning of this can be explained now in terms of definite but arbitrary numbers plugged into theplaces of x and y:

(5 + 3)2 = 52 + 2× 5× 3 + 32; (1729 + 691)2 = 17292 + 2× 1729× 691 + 6912; ....

It is hard not to be amazed that operations on numbers, once the height of abstraction, can now beconsidered as concrete interpretation.

Pythagoras (ca. 580–500 BC) is famous for the dictum that ‘all is number,’ possibly inspired bythe rational relationship between the lengths of strings that produce pleasant sounds when vibrating

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in unison. The scope of this example is ridiculously narrow when compared to the imagination of themodern physicist, who hesitates not to model all of reality in terms of far more complex superpositionsof vibrational modes. But Pythagoras thought not just of things in terms of numbers, but also numbersin terms of things. One finds reference to puzzling dictionaries of the following sort:

1=Number of reason;

2=First female number;

3=First male number;

4=The number of just revenge;

5=Marriage;

6=Creation;

and so on. In his Metaphysics, Aristotle appears to ascribe such a correspondence to conceptualimmaturity. That is, living in pre-Platonic times, Pythagoras is supposed still to have had greatdifficulty in thinking of a number its own, not in association with material things. Curiously enough,this is the plight of the modern arithmetic geometer as well, the enormous advances in human concep-tual sophistication notwithstanding. It is surprisingly helpful to think of numbers in terms of things,preferably spaces, when trying to solve algebraic equations in integers.

A case in point is the proof of Fermat’s last theorem. There, you’ll recall one starts from theobservation that it’s quite easy to add up two square numbers and find another square number as aresult:

32 + 42 = 9 + 16 = 25 = 52;

52 + 122 = 25 + 144 = 169 = 132;

82 + 152 = 64 + 225 = 289 = 172.

However, it turns out to be impossible to add up two (non-zero) cubes and obtain another cube, orto add up two fourth powers and obtain another fourth power. In general, the theorem states theimpossibility of finding non-zero triples of integers a, b and c satisfying the relation

an + bn = cn

for any n bigger than 2. In the elaborate proof, the beginning of the long story associates to such atriple of numbers a geometric object called an elliptic curve, which has the shape of a torus:

One then proceeds to show that the resulting torus has very special internal properties (dependingon an, bn, and cn), eventually rendering it an impossible object. Hence, the original triple must havebeen impossible.

What is equally impossible is to give you here any non-trivial sense of this procedure. Suffice it tosay that it is still remarkably useful to think of numbers in terms of visible objects.

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Nevertheless, on the whole, we are now pretty comfortable with recursive levels of abstraction1:

five penguins, five apples, five spoons - 5;

3, 5, 10, ..., - x.

And then, large segments of the population are able to thoughtlessly perform quite complex op-erations like multiplication and division with a facility that would have been utterly unimaginableto Pythagoras or Plato. The best mathematics proves its worth by becoming commonplace over theperiod of a few thousand years. It is an excellent thing that the question ‘What is number?’ is usuallyregarded as absurd these days.

After numbers were allowed independent existence, it seems many an entity struggled over theintervening millenia to be accorded the status of numberhood. You have certainly heard of theconvulsions surrounding zero, −1, or perhaps even of its imaginary square root. Our own age has seenthe ascendance of binary digits, like

10010111000

as one might find in some form inside a DVD or a message from the Cassini probe. Their initialoccurrence had very little to do with the general concept of a number, however, in that such digits weresimply regarded as a convenient form for representing words and images, reflecting well the mechanicsof the machines that used them. But now, we like to perform rather sophisticated operations on thesethings, such as

111 + 100 = 011,

or111× 100 = 001,

in coherent ways that must look utterly mysterious to outsiders.A child in a British primary school once responded to my query with the sensible opinion that a

number is something you can add and multiply. Indeed, without these operations, we may as wellhave written baababbbaaa instead of the long number above2. In any case, such strange operationshave become over the last few decades the most applicable portion of mathematics, essentially becauseof the advantage they provide for storing and correcting large quantities of data.

Frequently at the university, I teach a course on algebraic number theory, which, by and large, isconcerned with tightly structured rules for multiplying pairs or triples of integers in reasonable ways.For example, we might consider rules like

(a, b)(c, d) = (ac− bd, ad+ bc)

or(a, b)(c, d) = (ac− 10bd, ad+ bc).

For the second rule, concretely (!), we are saying something like

(2, 3)(5, 4) = (−110, 35).

What is the meaning of this? Theoretically, the reasons are similar to the situation of binary digits inthat the pairs become, thereby, algebraic structures in their own right rather than symbols (merelyidentifying a hotel room, for example). Here is a nice law of multiplication for triples of integers:

(a, b, c)(d, e, f) = (ad+ 13bf + 13ce, ae+ bd+ 13cf, af + be+ cd),

1Here, we indicate two. Even a moment’s reflection should convince you this is far from the whole picture. As atomsare the building blocks of molecules that come together to form cells, which, in turn, agglomerate into an organism, avariable of abstract algebra has in reality evolved through many layers of composite complexity.

2Some philosophers have undoubtedly noted the importance of this additional structure as well in the perennialdiscussion about essence. My ignorance prevents me from more respectable citation.

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and here is a case of it worked out with specific numbers

(1, 2, 3)(4, 5, 6) = (355, 247, 28).

A somewhat more complicated way to multiply is

(a, b, c)(d, e, f) = (ad+ 7bf + 7ce, ae+ bd+ 7cf, af + be+ cd).

The sense in which the first one is simpler is rather hard to explain in a few words, but I will remarkthat it yields a system much closer to the usual whole numbers. The key point is that the first systemhas a good notion of prime numbers similar to the familar one, while the second adds a layer ofcomplexity.

At the moment, such operations are probably known only to arithmeticians, interested to classifythem and derive consequences important to the study of a wide range of structures in pure mathe-matics. But I am fond sometimes of speculating about vast cities in space built in layers each oneabove another, wherein the typical house address will involve such triples of integers, long after theinefficient system of naming streets with words will have been forgotten. As the addresses are storedand transmitted in abundance along channels of futuristic communication, it is not hard to imagine ademand for good systems of multiplication that will facilitate the process. Eventually, a fluid enoughnotation for such multi-numerals will render them fully accessible to the public, and may appear abovedoorways in place of these unwieldy triples. I may be forgiven for predicting their acquisition at thelevel of primary school, in some form or another, three thousand years hence.

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The astute reader will have noticed already an obvious connection between number and space. Todescribe points in the plane, one needs two numbers (or a street name and a number), but for pointsin space, three numbers are natural. Pairs appear in the latitude and longitude for positions on theface of the earth, or when streets intersect,

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while a pilot in flight will be keenly aware of the altitude in addition to the coordinates of thepoint below him.

A methodology for keeping track of positions with collections of numbers in this way is called acoordinate system by learned people. Quite curiously, such a natural process of attaching numbers toobjects, a direction of correspondence opposite to that of Pythagoras, appears to have required a farmore precise command of scientific thinking than that of the ancient mystics. There are subtleties:you may have noticed with irritation, for example, that one cannot assign a longitude to the poles.That there can be no reasonable way to assign it can be seen by imagining yourself sitting right onthe south pole and observing all the longitudinal lines coming towards you. You will then see thatthere will be lines arbitrarily close to you with all possible longitudes. One can change the coordinatesin an attempt to avoid this. For example, a person living in Antarctica may well have covered thatcontinent with a nice grid similar to what we usually apply to points in the plane. Of course, thegrid will be cut off at some point, and the Antarctican might not particularly care about this anymore than a person in Manhattan minds that 125th street comes to an end. However, if he ends upexploring the seas beyond the boundaries of his world he will wish to extend his coordinates, probablyin a manner resembling the current system, but with different pairs of numbers attached to places.But the point is that no matter how he does this, he will always run into the same problem: his newsystem of number pairs cannot avoid collision somewhere (actually, at least two locations), and it willnever be possible to assign coordinates in a nice continuous fashion to the entire surface of the earth.In a similar vein, one necessarily makes an abrupt jump from positive to negative longitude whengoing around the the equator (or any other latitude) once.

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It is a quite general fact that the collection of numbers we use for describing locations on a geometricfigure will in some complex way reflect the shape under consideration. When properly systematizedinto an algebra of coordinates, these observations evolve into many different branches of geometry,eventually contributing to truly profound discoveries relating number and space. In three–dimensions,for example, it is a source of great mystery that there are portions of the universe, say the interior ofa black hole, that we cannot map in any way at all with numbers, even little bits at a time.

In the last few decades, we are gradually coming to the realization that arithmetic of any kind isconstrained by an implicit picture of shapes and their interactions informing the rules of operation.Consider, for example, the fact that

(A×B)× C = A× (B × C).

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These pictures are intended to convey a visual sense that that this algebraic property of multipli-cation is better modeled by the interaction of circles coming together in three-dimensional space thanthat of points colliding in the plane. The key point is that the transition from one bracketing to theother is visibly more smooth in three dimensions than in two.

In fact, we are currently witnessing the construction of a sophisticated algebra of space underlyingalmost all the algebraic structures of interest to mathematicians, and in which the precise nature ofmultiplication is the most important point to consider. Why multiplication? One reason comes fromthe study of physical systems, where the position of, say, a small particle in some multi-dimensionalspace might be approximately represented by a sequence of numbers like

ψ = (2, 3, 2, 5, 6, 8, 1, 9).

If there is another one in stateφ = (4, 9, 5, 3, 5, 8, 3, 1),

then the two particles in conjunction is specified, as if by black magic, using a multiplication ofparticles:

ψ⊗φ = (2×4, 2×9, 2×5, 2×3, 2×5, 2×8, 2×3, 2×1, 3×4, 3×9, 3×5, 3×3, 3×5, 3×8, 3×3, 3×1, . . .

. . . , 9× 4, 9× 9, 9× 5, 9× 3, 9× 5, 9× 8, 9× 3, 9× 1),

different again from any of those previously considered. This strange state of affairs might even consti-tute the ultimate origin of multiplication per se, used though we are to its being built out of additionin an elementary fashion for usual numbers. Here are some pictures illustrating a simple version ofspatial algebra, which I will leave you to contemplate for a moment without further explanation:

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My chief ambition as a young man reading science was to discover the basic constituents of spaceitself. What this could possibly mean I of course didn’t know at the time. It was something of anembarrassing curiosity that couldn’t have been divulged even to the sympathetic ear of a close friend.Perhaps I can attempt a few words now, having had a modicum of success in the classical path ofscholarship, whereby the gradual gathering of precision at the core of an inchoate question is thebeginning of discovery.

First let us note that space itself has a definite shape. To appreciate this, we need to consider for amoment that the evidence for the shape of anything at all is gleaned through a process of interactionand resistance. I may press against this table, for example, and deduce a contour from the gradualrealization that my hand will not move further in certain directions. Even without touching an object,our most immediate sense of shape is gathered from certain particles of light that experience similarresistance, a sequence of bounces along polygonal paths, before popping into our eye. What then isthe resistance of space? You may not think there is any, until you are asked anew why an attemptto jump off the ground under our feet will almost always end in a return. If I were to step out ofa spaceship somewhere near the orbit of Mars, but far away from the planet itself, I may well feelnothing at all, hence, a sensation of ‘floating.’ On the other hand, to my friends in the ship, I will beseen as trapped in a specific trajectory, quite likely terminating with a plunge into the sun.

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Thus, space appears to resist our pressing and flailing in a very persistent manner indeed. Amongthe several explanations possible, Einstein is famous for having proposed the shape of space itselfas the medium of this resistance, as the rocky slopes of a canyon might direct a stream along ameandering but well-defined path. Once we become fully aware that this shape is genuinely there,it becomes imperative to ask how it may be broken down into constituent pieces, as a building intobricks, granite into atoms, or a beam of light into the colors of a rainbow when passing through aprism. The answer will quite likely elude the best minds in the world for many years to come, andcertainly no such mystery is likely to yield to my own modest capabilities. Nevertheless, this is thequestion of questions that was uncomfortably present at the outset of my foray into the universe (ofscience), that led me to ponder the Pythagorean harmony (or struggle) of number and space, andeventually directs me to the algebra of space itself, of which the pale system of numbers making upquotidian concern is perhaps but a poor projection of a shadow onto the rough walls of a cave. Passingnow the middle of my life, I aspire to return at some point to this beginning.

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...With equal passion I have sought knowledge. I have wished to understand the heartsof men. I have wished to know why the stars shine. And I have tried to apprehend thePythagorean power by which number holds sway above the flux. . . Love and knowledge,so far as they were possible, led upward toward the heavens. But always pity brought meback to earth. Echoes of cries of pain reverberate in my heart. Children in famine, victimstortured by oppressors, helpless old people a burden to their sons, and the whole world ofloneliness, poverty, and pain make a mockery of what human life should be...(Bertrand Russell, autobiography)

Who among us has not burnt with this passion or suffered not from like despair? If I were to beroundly scolded for such self-indulgent distractions as the nature of number or the atoms of space,and told instead to till the fields or sweep rubbish in public avenues, perhaps my heart would knowno rejoinder. But enough of humanity, faithful souls like the late Lee Won-Kook, tolerate our glassypursuits and even help us along in them with generous gifts built up from the kind of hard work of alifetime that puts my daily excesses to complete shame. Since I am not good with ceremonious words,I may be forgiven a small personal anecdote to help convey my thanks. My son started secondaryschool on Friday of last week and, as may be common among boys of that age, expressed trepidationand dislike around a new beginning. Since this lecture was on my mind, I came to tell him about thegift of Mr. Lee, from which I benefit and he as well through me. I supposed I should have urged himto count his blessings or some such thing. But I have never been a good father in that way, able toimpart the proper lessons of moral sensibility in a convincing manner. Instead, I think the point of mysermon was the wonder of it all, the good will, action, and sense of purpose that manages to spreadfrom one person to another through incidental encounters and infect the world like an anti-disease.A relevant story comes from my own father, who is a literary man himself, but who has urged me tostudy science starting at a young age. He remembered his science teacher from a primary school inGwangju, gifted in the classroom, but endowed also with the kind of gentle passion for knowledge thatendears in the midst of instructing. In those slower times, he would oftentimes gather the children ona hilltop near school in the dark of night to explain the planets and the galaxy, the mythology of theconstellations. I heard this story around the time I was finishing university, and the conjecture thatthis teacher was probably the reason I had ended up a scientist. Thus, A teaches B, who influencesD, E, F, slightly or profoundly. They may then come together to do something useful or truthful, orone or two may have children, who then, god willing, will carry with them some bodily memory ofthe good that went through A. The wonderful environment for study and research one finds at thisuniversity through the generosity of people like Mr. Lee is certainly reason enough for deep gratitude.But perhaps even more comforting on this day is our remembrance of that organic algebra of thefaithful, extending its vast operations across generations and eons, keeping our migrant souls turnedtowards dim horizons of meaning and truth, mindful even amid our illusions of solitude.

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